Transcript Ch 6 - csusm

Chapter 6
Sample mean
 In statistics, we are often interested in finding
the population mean (µ):
 Average Household Income in San Diego Area
 Average Student’s SAT score in CSUSM
 Average hours per week a person spend on the
beach
 Average temperature in San Diego
 Most of the time, we only study a sample 
sample mean (x).
 Collecting information for the population may be
too costly or just impossible.
 We cannot get the exact information, but an
estimation.
 The estimation varies depending the actual
sample we get.
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A simple example to
observe:
 Assume the population is composed of
the following 4 people:
 A: John, age =26
A
B
C
D
 B: Mary, age = 24
 C: Cindy, age = 22
 D: Mike, age = 28
μ=?
σ=?
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Sample:
 Use random sampling technique
(with replacement) to select 2 of
them.
 If Cindy and John were selected
 What is the mean age for the
sample?
 John: 26 and Cindy: 22
 Sample mean?
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Sampling Error
sampling error = x  
 For example, the sampling error in the above
example is 24 – 25 = -1
 Note the order! Sample mean first, population
mean second.
 The sampling error has a sign.
 “+” means the sample mean is too high
 “-” means the sample mean is too low.
 What is the probability to get a positive sampling
error? probability to get a negative one?
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Example
 Problem 6.4 (page 234, in
homework)
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Understanding Sampling
error
 Why does sampling error exist?
 Because you only examined a part of
the population to get the sample
mean
 What if sample size gets larger?
 Sample mean should get closer to
population mean
(The more you sampled, the more
reliable your sample mean is as an
estimation of population.)
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Exercise
 Problem 6.1 and 6.8
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Sample mean
 Sample mean may not has the same
value as population mean.
 Sampling errors always exist.
 Sample mean varies when you get
different samples.
 Before you sample, the sample mean
has uncertain value
 It can be treated as a “random variable”
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The Distribution of sample
mean
 Sample mean is a random variable
(before Sampling).
 Hence, it has a distribution.
 What should be the mean of this distribution?
 How will the variance of this distribution change
if the sample size gets larger?
 What does this distribution look like?
 Symmetric? Skewed?
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Distribution of
sample mean
Example: In the population of the four people
(assume the selection is with replacement.
That is, the two people being selected could
be the same)
The second person selected
The first person
selected
John
(26)
Mary
(24)
Cindy
(22)
Mike
(28)
John
(26)
Mary
(24)
Cindy
(22)
Mike
(28)
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Probability Distribution
 What is a probability distribution?
 How many ways to present a
probability distribution?
 How to assign probabilities?
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Central Limit Theorem
Theorem: IF
(1) the population has mean  and
variance 2, and
(2)the sample size n is large enough,
THEN: the sample mean x follows a
normal distribution,
mean: μ x =μ
2
σ
variance: σ 2x =
n
σ2
σ
standard deviation: σ x =

n
n
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Example
Assuming a sample of 50 is selected from
a population. The population mean is 500,
and variance is 25, determine the mean,
variance, and the standard deviation of
sample mean.
sample size: n=50
population mean: μ=500
population variance:  2  25
μ x =μ=500
2
σ
25
2
σx =

=0.5
n
50
σ x = σ 2x = 0.5  0.707
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Comments on the CLT
 Two scenarios:
1. The population is normal:
sample size doesn’t have to be large to
guarantee the sample mean distribution.
2. The population is not normal
sample size has to be large. The larger the
better!
The theorem is just an approximation.
 The expected value (mean) of sample
mean is always the population mean

What does it mean?
 The variance of the distribution is small
when


the population variance itself is small; or
the sample size is large.
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Concepts clarification
 In chapter 3 (numerical measures), we
compute the following given a set of
sample data
 Sample mean:
x
 Sample variance:
s
sample
statistics
2
 Sample standard deviation:
s
 In this chapter, we are examining the
characteristics of sample mean
x
 It is a random variable
 The mean of the sample mean μ x
 The variance of the sample mean: σ 2x
 The standard deviation of sample mean: σ x
They are population parameters!
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Determine the probability
 What is the probability a sample mean
falls into a certain range?
e.g. P(3<x<5)?
1. Sample mean is normally distributed!
2. In order to figure out the probability, we
should use normal table
3. In order to use normal table, we need to
know the z scores…
4. In order to know the z-scores, we need to
know the mean and standard deviation of
this normal distribution
5. Do we? Yes, we can!!

Population parameters (, 2)

Sample size n

Central Limit Theorem (CLT)
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Exercise
 Problem 6.23 (a) P249
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Summary
 In this chapter, we studied
Sample mean
 Sample mean varies based on
different sample.
 It can be regarded as a random
variable
 When the sample size is large
enough, it has a mean of  and
standard deviation /n.
 We can compute the probability
that the sample mean falls in a
range using normal table.
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